Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 “Cardinality Equals Variety for Chords” in Well-Formed Scales, with a Note on the Twin Primes Conjecture
- 2 Flip-Flop Circles and Their Groups
- 3 Pitch-Time Analogies and Transformations in Bartók's Sonata for Two Pianos and Percussion
- 4 Filtered Point-Symmetry and Dynamical Voice-Leading
- 5 The “Over-Determined” Triad as a Source of Discord: Nascent Groups and the Emergent Chromatic Tonality in Nineteenth-Century German Harmonic Theory
- 6 Signature Transformations
- 7 Some Pedagogical Implications of Diatonic and Neo-Riemannian Theory
- 8 A Parsimony Metric for Diatonic Sequences
- 9 Transformational Considerations in Schoenberg’s Opus 23, Number 3
- 10 Transformational Etudes:Basic Principles and Applications of Interval String Theory
- Works Cited
- List of Contributors
- Index
- Miscellaneous Frontmatter
4 - Filtered Point-Symmetry and Dynamical Voice-Leading
Published online by Cambridge University Press: 10 March 2023
- Frontmatter
- Contents
- Preface
- Introduction
- 1 “Cardinality Equals Variety for Chords” in Well-Formed Scales, with a Note on the Twin Primes Conjecture
- 2 Flip-Flop Circles and Their Groups
- 3 Pitch-Time Analogies and Transformations in Bartók's Sonata for Two Pianos and Percussion
- 4 Filtered Point-Symmetry and Dynamical Voice-Leading
- 5 The “Over-Determined” Triad as a Source of Discord: Nascent Groups and the Emergent Chromatic Tonality in Nineteenth-Century German Harmonic Theory
- 6 Signature Transformations
- 7 Some Pedagogical Implications of Diatonic and Neo-Riemannian Theory
- 8 A Parsimony Metric for Diatonic Sequences
- 9 Transformational Considerations in Schoenberg’s Opus 23, Number 3
- 10 Transformational Etudes:Basic Principles and Applications of Interval String Theory
- Works Cited
- List of Contributors
- Index
- Miscellaneous Frontmatter
Summary
Introduction
Until now, diatonic systems and neo-Riemannian transformations have generally been considered separately, and group and graph theoretic approaches have dominated the neo-Riemannian and transformational theory literature. In this paper, an alternative approach will be explored; techniques similar to those used in the study of dynamical systems in science will be adopted to study neo- Riemannian theory and its connection to diatonic theory.
Dynamical systems are probably best known today for the fractals they sometimes generate (e.g., Koch's snowflake, the Dragon curve, Mandelbrot's set, the Julia set, etc.), but fractals are only part of this field of study. As Strogatz puts it in his text on nonlinear dynamics and chaos, dynamics ”… is the subject that deals with change, with systems that evolve in time. Whether the system in question settles down to equilibrium, keeps repeating in cycles, or does something more complicated, it is dynamics we use to analyze the behavior.”
Dynamical systems related to the topics that will be discussed here will utilize point-symmetric concentric circles that rotate through time and stroboscopic portraits, which record events at particular time intervals. These dynamical systems can be thought of as sequence generators which, with the appropriate choice of control parameters, produce periodic orbits (cycles) of scales and chords well known in both diatonic and neo-Riemannian theory.
Douthett and Steinbach's Relation Definition will also be used:
Let X and Y be pcsets. Then X and Y are Pm,n-related if there exists a set {xk}m+n+1k=0 and a bijection τ : X →Y such that X \ Y (the set of pcs in X that are not in Y ) ={xk}m+n+1k=0, τ(x ) = x if x ∊X ∩Y , and
The requirement that τ be a bijection implies that X and Y have the same cardinality.
- Type
- Chapter
- Information
- Music Theory and MathematicsChords, Collections, and Transformations, pp. 72 - 106Publisher: Boydell & BrewerPrint publication year: 2008
- 27
- Cited by